Abstract:

Tensor product function (TPF) approximations are widely used to solve high-dimensional problems, such as partial differential equations and eigenvalue problems, achieving remarkable accuracy with computational costs that scale linearly with problem dimensions. However, recent studies have highlighted the prohibitively high computational cost of TPFs in quantum many-body problems, even for systems with as few as three particles. A key factor contributing to this challenge is the antisymmetry requirement imposed on the unknown functions.In this work, we rigorously demonstrate that the minimum number of terms required for a class of TPFs to satisfy exact antisymmetry grows exponentially with the problem dimension. This class includes both traditionally discretized TPFs and those parameterized by neural networks. By establishing a connection between antisymmetric TPFs and their corresponding antisymmetric tensors, we analyze the Canonical Polyadic rank of the latter to derive our results.Our findings reveal a fundamental incompatibility between antisymmetry and low-rank TPFs in high-dimensional settings. This work provides new insights into the limitations of TPFs and offers guidance for future developments in this area.

Biography：

刘歆，中国科学院数学与系统科学研究院“冯康首席研究员”，博士生导师，计算数学与科学工程计算研究所副所长。

刘歆2004年本科毕业于北京大学数学科学学院；并于2009年在中国科学院数学与系统科学研究院获得博士学位。主要研究方向包括流形优化、分布式优化及其在材料计算、大数据分析和机器学习等领域的应用。刘歆分别于2016年，2021年和2023年获得国家自然科学基金委优秀青年科学基金项目、杰出青年科学基金项目和科技部重点专项的资助。2024年获得中国工业与应用数学学会萧树铁应用数学奖。现担任MPC, JCM, JIMO, APJOR等国内外期刊编委，《中国科学·数学》（中英文）青年编委，《计算数学》副主编；中国科学院青年创新促进会理事长；中国运筹学会常务理事；中国工业与应用数学会副秘书长，中国数学会计算数学分会常务理事。